#349178

order	512
nilpotency class	3
isoclinism	isoclinic class of #349150
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	D₄×#41	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,7)(3,5,9,15)(4,11,13,10)(8,12,16,14)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,3,5,7,6,9,15,2)(4,10,12,16,13,11,14,8)(17,19)(18,22)(20,27)(21,28)(23,30)(24,26)(25,31)(29,32);(2,7)(3,9)(8,16)(10,11)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,4)(2,8)(3,10)(5,12)(6,13)(7,16)(9,11)(14,15)(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=di,f^c=fi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256374);